nLab
partial trace

Suppose V, W are finite-dimensional vector spaces over a field, with dimensions m and n, respectively. For any space A let L(A) denote the space of linear operators on A. The partial trace over W, TrW, is a mapping

TL(VW)Tr W(T)L(V).T \in L(V \otimes W) \mapsto Tr_{W}(T) \in L(V).
Definition

Let e 1,,e m and f 1,,f n be bases for V and W respectively. Then T has a matrix representation {a kl,ij} where 1k,im and 1l,jn relative to the basis of the space VW given by e kf l. Consider the sum

b k,i= j=1 na kj,ijb_{k,i} = \sum_{j=1}^{n}a_{kj,ij}

for k,i over 1,,m. This gives the matrix b k,i. The associated linear operator on V is independent of the choice of bases and is defined as the partial trace.

Example

Consider a quantum system, ρ, in the presence of an environment, ρ env. Consider what is known in quantum information theory as the CNOT gate:

U=0000+0101+1110+1011.U={|00\rangle}{\langle 00|} + {|01\rangle}{\langle 01|} + {|11\rangle}{\langle 10|} + {|10\rangle}{\langle 11|}.

Suppose our system has the simple state 11 and the environment has the simple state 00. Then ρρ env=1010. In the quantum operation formalism we have

T(ρ)=12Tr envU(ρρ env)U =12Tr env(1010+1111)=1100+11112=11T(\rho) = \frac{1}{2}Tr_{env}U(\rho \otimes \rho_{env})U^{\dagger} = \frac{1}{2}Tr_{env}({|10\rangle}{\langle 10|} + {|11\rangle}{\langle 11|}) = \frac{{|1\rangle}{\langle 1|}{\langle 0|0\rangle} + {|1\rangle}{\langle 1|}{\langle 1|1\rangle}}{2} = {|1\rangle}{\langle 1|}

where we inserted the normalization factor 12.

Revised on March 6, 2010 23:40:54 by Toby Bartels (75.117.106.207)